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have added a tad more content to Stein manifold and cross-linked a bit more
Added the statement of Cartan’s theorem B and added to the Idea-section a remark that therefore Stein manifolds play the role in complex geometry of Cartesian spaces in smooth manifold theory, for purposes of abelian (Cech)-sheaf cohomology.
Affine schemes of algebraic geometry.
Or close at least. I have added something here.
I am not saying that (about the analytification), but precisely what you say above: affine are cohomologically trivial in the sense as proved in chapter 3 of Hartshorne’s book and this is a usually given statement when algebraic geometers look at analytic spaces. Many other deep properties are also analogous.
somebody just alerted me:
this page here has been and still is referring to
for proof of some of its statements (existence of “good” Stein covers). However, the link to that pdf
http://www.math.columbia.edu/~maddockz/notes/dolbeault.pdf
no longer works, and Google seems to see no other trace of it either.
(?)
Zachary Maddock is on Linkedin so you might be able to contact him and put a copy of the document on the Lab if it seems worth it.
For what it’s worth, I have found and uploaded an old copy of the file (here)
adjusted the wording of the example of open Riemann surfaces (here) and added pointer to the classical reference:
adjusted the wording of the example of open Riemann surfaces (here) and added pointer to the classical reference:
Just a sanity check: are disjoint unions of Stein manifolds again Stein manifolds? Including countably infinite disjoint unions? The answer seems to be yes, simply by staring at the definition, but perhaps I am missing something?
The reason I am asking about this is that Lárusson in his paper https://arxiv.org/abs/math/0101103v3, at the beginning of Section 5 asks “It is natural to ask whether a finite homotopy sheaf on S satisfies descent.”
Here S is the site of Stein manifolds and holomorphic maps, and a finite homotopy sheaf is a presheaf that satisfies the homotopy descent condition with respect to finite covers.
It would seem to me that the answer to Lárusson’s question should be negative as stated, e.g., because we can take the presheaf P that assigns to a Stein manifold M the abelian group of holomorphic functions on M that vanish on all but finitely many connected components of M.
Then P satisfies descent with respect to finite covers, i.e., is a finite homotopy sheaf in Lárusson’s terminology. However, it does not satisfy descent with respect to (say) countable disjoint covers.
I don’t see why not. Use a characterisation of n-dimensional Stein manifolds that involves proper embeddings in C^m for some m. In fact there is a uniform bound on the needed m in terms of n, so for countable disjoint unions there is a common m. Then properly embed embed the countable copies of C^m in C^m+1 as parallel affine hypersurfaces.
Or else I can ask Finnur today, as his office is next to mine, if that’s not convincing enough. Why do you ask?
Re #12: Basically, I am wondering why the example in the last two paragraphs of #11 is not a (trivial) counterexample to what Lárusson is suggesting in his paper https://arxiv.org/abs/math/0101103v3, at the beginning of Section 5, where he asks “It is natural to ask whether a finite homotopy sheaf on S satisfies descent.”
I should add that one doesn’t need all the dimensions of the disjoint pieces to be equal, but you do need a bound on the dimensions. I spoke with Finnur today and he seemed to indicate that if the dimensions of the pieces are unbounded, then the disjoint union isn’t Stein. Maybe this is it.
if the dimensions of the pieces are unbounded, then the disjoint union isn’t Stein
That’s not the example I had in mind, though.
Disjoint unions of points (finite or countable) are 0-dimensional Stein manifolds.
Finitely supported complex-valued functions on such manifolds form a presheaf that satisfies finite homotopy descent, but does not satisfy descent with respect to covers of arbitrary cardinality. This would seem to constitute a simple counterexample to the question stated in Section 5 of his paper, but probably I am just misreading something.
Oh, sorry, I was kinda ignoring the homotopy sheaf material :-)
Dmitri’s point in #11 (repeated in #13) isn’t even to do with homotopy, it’s a rather basic observation about sites and sheaves.
Dmitri brought up a possible subtlety in the definition of Stein manifolds only as an attempt to find a technical loophole clause that would explain why Lárusson doesn’t consider what seems to be the immediate conclusion.
For when the editing functionality is back, to add the example that complements of hyperplane arrangements in Stein manifolds are again Stein (e.g. the configuration spaces of points in a punctured Riemann surface are Stein).
This is:
recalled, e.g., in
Also to add pointers for this example: Universal covers of Stein manifolds are again Stein:
recalled, e.g., in:
This monograph cares to spell out the proof that homolorphic de Rham cohomology of Stein manifolds computes ordinary cohomology:
review and generalization in:
Also good to mention: The intersection of any finite number of open Stein subsets inside some complex manifold (which itself need not be Stein) is again Stein:
From this and the usual facts about the holomorphic de Rham complex (e.g. as usefully reviewed by M. Stevenson here: pdf), it is immediate that the the holomorphic De Rham cmplex satisfies descent when regarded as an $\infty$-stack over the site of Stein manifolds (i.e. any shift of it, seen under the Dold-Kan construction).
I wonder if any author already says this, in a citable way?
Is there a citable textbook that would make explicit the fact that over Stein domains the global sections of a twisted holomorphic de Rham complex computes the cohomology in the given local system?
several authors who discuss related issues don’t mention this particular combination of the statement (i.e. holomorphic + twists + Stein).
For instance Voisin’s book (translated by Schneps), where this is part of Cor. 5.4 (here) the issue seems to be a little lost in translation (“varieties” on p. 131 remains ambiguous).
made more explicit (here) that on Stein manifolds holomorphic de Rham cohomology coincides with ordinary de Rham cohomology.
I gather that this statement may originate with
though I haven’t actually seen this article, since the DOI (which I get from SemanticScholar here) seems to be broken.
But the same statement appears then also in
which I have added, too.
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